Independence of $\ell$ for the supports in the Decomposition Theorem

TL;DR

This paper proves that the supports of irreducible perverse sheaves in the Decomposition Theorem are independent of the prime number , extending Gabber's result to a relative setting.

Contribution

It generalizes Gabber's -independence result for intersection cohomology to the context of the Decomposition Theorem with supports and local systems.

Findings

Supports of irreducible perverse sheaves are -independent

Family of local systems on each support is -independent

Extends Gabber's -independence to the relative case

Abstract

In this note, we prove a result on the independence of $\ell$ for the supports of irreducible perverse sheaves occurring in the Decomposition Theorem, as well as for the family of local systems on each support. It generalizes Gabber's result on the independence of $\ell$ of intersection cohomology to the relative case.